From Longman Dictionary of Contemporary Englishthink something ↔ up phrasal verbINVENTto produce a new idea, name etc by thinking She was trying to think up an excuse. Did you think that up yourself? Who thinks up names for new products? → think→ See Verb table
Examples from the Corpus
think up• How do they think them up?• Not once did she think of giving up.• Today I would like you to think about taking up a new interest or hobby.• And I am not really sure of it at all, she thought, standing up and looking about her.• At last! she thought, looking up at John.• Lately I have been thinking of taking up cigarettes again.• I think they put up forty-four units in one place, and about fifty units in an-other.• I think that shows up in certain issues. 